Codes inside shells of low-dimensional integer lattices

  • Mikhail Ganzhinov (Creator)



Let S(n,t,k) be the maximum size of a code containing only vectors of the kth shell of the n-dimensional integer lattice such that the inner product between distinct vectors does not exceed t. The file A.pdf contains examples of codes leading to lower bounds for S(n,t,k), one example for the each lower bound. By (n,k,t,N) we denote a code containing N vectors of the kth shell of the n-dimensional integer lattice such that the inner product between any two different vectors is at most t. The following format to describe example codes is used: — If a code is constructed using a signed permutation group, then a list containing generators of the corresponding group G is given first followed by a list containing used block system and G-orbits forming the code. Each orbit is represented by a list containing orbit representative and an orbit size (in that order). Finally, a code may also contain explicitly added vectors. In this case corresponding list contains two items: a number of explicitly added vectors and an another list containing the vectors. — If a permutation group is used, then the format stays the same except the list containing block system is not present. — Some codes are constructed by scaling spherical codes obtained by Y_n constructions described in [1]. If one part of an example code is constructed by using a signed permutation group, then Y_n-constructed part also utilizes the same block system to describe components of codevectors. — Notation W(n,w) is used to denote a weighing matrix of order n and weight w used to map vectors of a (n,k,t,N)-code into kwth shell of n-dimensional integer lattice to produce a (n,kw,tw,N)-code. The file B.pdf lists some examples of large spherical codes containing vectors from multiple shells. To obtain spherical codes scale given vectors appropiately. If only parameters of a code are given without the construction, then this code is one of the examples given in A.pdf. [1] Ericsson, T.; Zinoviev, V.: Codes on Euclidean Spheres. Elsevier Science (2001).
Date made available6 Mar 2024

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